Statistical theory of Ising model and Heisenberg model

Horaƫiu Năstase

Yu. G. Rudoi

Bernhard M�hlschlegel, H. Zittartz

Rainer J. Jelitto

Abstract By pushing forward the decoupling from the three-to the four-point correlation function short-range order is systematically introduced into the description of the statistical behaviour of the Ising-model. Application of a procedure which is a natural generalization of that invented by Bogoljubov and Tjablikov for the Heisenberg-model, leads to an overall-approximation for the magnetization and the nearest-neighbour correlation which may be compared with the Ising-model variant of Oguchi's two-spin-cluster molecular field theory. The results of both approximations are very similar both for low and high temperatures, but for the transition point the new approach yields values which lie considerably lower and therefore are more reliable than those following from Oguchi's theory. Moreover, the comparison with a slight modification of the theory which is also presented in this paper, illuminates the physical mechanism which is responsible for the formation of correlations within the order of approximation, considered.

Walter Greiner, Ludwig Neise, Horst Stöcker B

A.N. Vasiliev

This chapter is devoted exclusively to systems in equilibrium. First it considers the quantum statistics of systems like the Bose or Fermi gas, then it considers lattice spin systems and the nonideal classical gas. Quantum statistics is Euclidean field theory on a finite time interval with the additional condition that the fields be periodic. Functional integrals in statistics differ from the integrals of Euclidean field theory only by the finiteness of the range of integration over time in the action functional and the periodicity requirements on the fields. The classical Heisenberg model differs from the Ising model in that with each lattice site is associated a classical d -dimensional spin vector s of unit length. The chapter also discusses the case of a gas with many-body forces which is interesting mainly because it leads to a diagrammatic technique which lies outside ordinary graph theory.